Superconductivity, broken gauge symmetry, and the Higgs mechanism

Nicholas R. Poniatowski

Citation: American Journal of Physics 87, 436 (2019); doi: 10.1119/1.5093291

View online: https://doi.org/10.1119/1.5093291
View Table of Contents: https://aapt.scitation.org/toc/ajp/87/6
Published by the American Association of Physics Teachers

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Superconductivity, broken gauge symmetry, and the Higgs mechanism
Nicholas R. Poniatowskia)
Department of Physics, Center for Nanophysics and Advanced Materials, University of Maryland, College
Park, Maryland 20742
(Received 11 September 2018; accepted 9 February 2019)
The association of broken symmetries with phase transitions is ubiquitous in condensed matter
physics: crystals break translational symmetry, magnets break rotational symmetry, and
superconductors break gauge symmetry. However, despite the frequency with which it is made,
this last statement is a paradox. A gauge symmetry, in this case the U(1) gauge symmetry of
electromagnetism, is a redundancy in our description of nature, so the notion of breaking such a
“symmetry” is unphysical. Here, we will discuss how gauge symmetry breaks, and doesn’t, inside a
superconductor, and explore the fundamental relationship between gauge invariance and the
striking phenomena observed in superconductors. VC 2019 American Association of Physics Teachers.
https://doi.org/10.1119/1.5093291

I. INTRODUCTION
Here and throughout the remainder of the paper, we will use
If one spends enough time running in condensed matter natural units where h ¼ c ¼ 1, and further choose
circles, one will surely hear the off-handed remark that super- e0 ¼ l0 ¼ 1. The second and third equations may be satisfied
conductors break gauge symmetry. On the other hand, our by writing the physical E and B fields in terms of the electro-
undergraduate electromagnetism class taught us that a gauge magnetic potentials
symmetry is a mere redundancy, not a physical symmetry that
can reasonably be “broken” in a consistent theory. So, what @A
does it actually mean for a gauge symmetry to “break,” and E ¼ ru  ; (5)
@t
why is this tied to superconductivity? In this article, we will
address these questions, paying special attention to the role B ¼ r  A: (6)
the mean field approximation plays in the study of supercon-
ductivity, and demonstrate how a classical field theory with However, the choice of u and A is not unique for a given
spontaneously broken gauge symmetry can account for many configuration of E and B. Since r  (ra) ¼ 0 for any scalar
of the spectacular properties of superconductors. field a(x, t), the transformation A 7! A þ ra leaves the mag-
The first four sections include preliminary information, netic field B ¼ r  A unchanged. To ensure the electric
enabling a reader with only a basic familiarity with quantum field is also invariant, we must simultaneously transform the
mechanics to follow the remainder of the discussion. We scalar potential as u 7! u  @t a. This intrinsic ambiguity in
review what precisely electromagnetic gauge symmetry is in the electromagnetic potentials is called gauge symmetry, and
Sec. II, the notion of spontaneously broken symmetries in the transformation
Sec. III, and introduce second quantization and elements of
quantum field theory in the context of superconductivity in A 7! A þ ra;
@a (7)
Sec. IV. The reader familiar with this material is encouraged u 7! u  ;
to skip ahead to the main discussion of spontaneous gauge @t
symmetry breaking inside superconductors which comprises
under which the electromagnetic field is invariant is called a
the remainder of the article. Specifically, we will discuss the
gauge transformation. Since our choice of the gauge parame-
constraints gauge invariance imposes on any effective
ter a is arbitrary, all physical observables must be indepen-
description of the superconducting state following Refs. 1
dent of a, that is, gauge invariant.
and 2, and finally carefully consider the Anderson-Higgs
To incorporate quantum mechanics, recall that in the pres-
mechanism which is the crux of the gauge-symmetry-breaking
ence of an electromagnetic field the momentum of a particle
story.
with charge q must be shifted4 p 7! p  qA which corre-
sponds to the replacing the operator ihr 7!  ihr  qA in
II. ELECTROMAGNETIC GAUGE SYMMETRY quantum mechanics. The Schr€odinger equation for such a
Let us review how gauge symmetry arises in classical charged particle in a potential V(x) in the presence of an elec-
electromagnetism,3 starting with the Maxwell equations tromagnetic field is5

r  E ¼ q; (1) @w 1
i ¼ ðr  iqAÞ2 w þ ðV þ quÞw: (8)
@t 2m
r  B ¼ 0; (2)
@B If we gauge transform the fields while locally rotating the
rEþ ¼ 0; (3) phase of the wave function by the gauge parameter evaluated
@t
at each point in space and time
@E
rB¼jþ : (4)
@t wðx; tÞ 7! eiqaðx;tÞ wðx; tÞ; (9)

436 Am. J. Phys. 87 (6), June 2019 http://aapt.org/ajp C 2019 American Association of Physics Teachers
V 436
the Schr€ odinger equation becomes6 magnetization over any mesoscopic distance. In contrast, the
  ferromagnetic phase has a finite magnetization. Thus, the
@ @a 1 magnetization per unit volume m(x) may serve as an order
i þ iq w¼ ðr þ iqra  iqA  iqraÞ2 w
@t @t 2m parameter. To identify the symmetry broken at the transition,
  we note the system is isotropic in the paramagnetic phase,
@a
þ V þ qu  q w: but the ferromagnet is magnetized along a particular direc-
@t tion in space. Thus, the ferromagnet spontaneously breaks
(10) rotational symmetry by “choosing” a magnetization axis.8
Note however, that the magnetization axis is arbitrary.
The a dependent terms drop out, and we recover the same Thus, there is in fact a continuum of degenerate ground
Schr€
odinger equation we started with. Thus, the Schr€odinger states: one corresponding to each possible magnetization
the Schr€
odinger equation is gauge invariant under the direction. Furthermore, it costs no energy to uniformly
quantum gauge transformation rotate the system from one degenerate ground state into
@a another. Then, by continuity, we expect that arbitrarily long
wðx; tÞ 7! eiqaðx;tÞ wðx; tÞ; A 7! A þ ra; u 7! u  : wavelength variations in the magnetization direction will
@t have an arbitrarily small energy cost. In the ferromagnet,
(11) these variations, called spin waves, dominate the low
energy excitations of the system since they are gapless
Due to the phase rotation, this gauge symmetry is also said (their energy can be made vanishingly small). In fact, a
to be a local U(1) symmetry. It is local because we may general consequence of spontaneously broken continuous
rotate the phase of the wave-function by a different amount symmetries is the emergence of gapless excitations around
at every point in space, in contrast to the weaker global sym- the ground state called Goldstone modes,7 of which the spin
metry where we may only rotate the phase of the wave- wave is an example.
function by the same fixed amount at every point. Further,
U(1) is the group of one-dimensional unitary matrices, which
IV. SUPERCONDUCTIVITY
is equivalent to the set of all phases {eih} by which we may
multiply the wave function. As in the classical case, all phys- A superconductor, as the reader is likely aware, is a phase
ically meaningful, observable quantities must be gauge of matter exhibiting a number of extreme phenomena,
invariant under the transformation in Eq. (11) due to the arbi- including the eponymous perfect conductivity and the expul-
trariness of the gauge parameter. sion of magnetic fields, or Meissner Effect. Microscopically,
electrons are bound via an attractive interaction into Cooper
III. SPONTANEOUS SYMMETRY BREAKING pairs, which condense into a common macroscopic quantum
state.
Roughly speaking, a symmetry of a particular theory is a Since superconductivity is an inherently quantum phe-
transformation which leaves the physics unchanged. More pre- nomenon as well as a many-body effect, it is most naturally
cisely, it is a transformation under which the Hamiltonian or described in the language of second quantization. We very
Lagrangian defining the theory is invariant. We can explicitly briefly review this formalism in Sec. IV A, but the familiar
break this symmetry by adding terms to the Hamiltonian or reader is encouraged to skip this cursory treatment.
Lagrangian which do not respect the symmetry, corresponding
to an external perturbation of the system. More interesting is
A. Second quantization
the case of spontaneous symmetry breaking, in which the sys-
tem finds itself in a state that does not respect the symmetry of Recall the harmonic oscillator from elementary quantum
the underlying Hamiltonian or Lagrangian.7 For the purposes mechanics, the Hamiltonian of which may be “factored” by
of this paper, we will say a symmetry is spontaneously broken introducing the operators
when the ground state of a system is not invariant under a sym-
metry of the Hamiltonian or Lagrangian. 1
This notion of symmetry breaking is central to the para- a ¼ pffiffiffiffiffiffiffiffiffiffi ðmxx þ ipÞ; a† ¼ h:c:; (12)
2mx
digmatic Landau theory of phase transitions. Since con-
densed matter systems are extraordinarily complex, which are interpreted as raising and lowering the system
diagonalizing the microscopic Hamiltonian for any interact- between energy eigenstates jni, such that (neglecting nor-
ing system is typically a hopeless endeavor. Instead, the malizations), a† jni  jn þ 1i and ajni  jn  1i. Further,
Landau picture involves using the symmetries of the system these eigenstates are orthonormal
to deduce its long distance behavior near a phase transition.
In particular, we identify a symmetry respected by the phase
on one side of the transition, and broken by the other (invari- hnjmi ¼ dnm : (13)
ably, the ordered phase is less symmetric). Here, the symme-
try we speak of is meant in the coarsest sense, since we have Now, suppose we wish to consider a system that does not
not introduced any Hamiltonian to appeal to. Instead, we conserve particle number, i.e., one in which particles may be
quantify the symmetry by identifying an order parameter: an created or destroyed. We can introduce analogous creation
observable whose expectation value vanishes in the symmet- and annihilation operators
ric phase and is nonzero in the symmetry breaking phase.
To be concrete, consider a ferromagnet which, below the W† ðxÞ creates particle at point x; (14)
system’s Curie temperature Tc, spontaneously magnetizes.
Above Tc, the system is a paramagnet, with no net WðxÞ annihilates particle at point x: (15)

437 Am. J. Phys., Vol. 87, No. 6, June 2019 Nicholas R. Poniatowski 437
 Y
In allowing for the particle number of our system to change, jXBCS i ¼ ðvk þ uk c†k" c†k# Þj0i; (21)
our space of possible states has expanded from a Hilbert k
space to a Fock space, which can be thought of as collection
of “sectors” corresponding to a particular particle number, where c†kr is the momentum space electron creation operator.
each of which is itself a Hilbert space.9 Denoting an N-particle Given the second term, we evidently have a state of indefi-
state as ja; Ni where the index a represents all other quantum nite particle number.
numbers, we may write Having seen the correlation function hW# W" i is zero in the
normal phase and nonzero in the superconducting phase, it
W† ja; Ni ¼ jb; N þ 1i; (16) may serve as an order parameter for the superconducting
state.14 Following BCS, we define the superconducting order
Wja; Ni ¼ jb; N  1i; (17)
parameter
ha; Njb; Mi ¼ hajbi dNM : (18)
D / hW# W" i; (22)
So we see the creation and annihilation operators take a state
from one sector of the Fock space into another, and that which we will later see coincides with the previously defined
states in different sectors of Fock space are orthogonal,10 condensate field also labeled D.
which is fairly intuitive since the particle number can only Under a gauge transformation, creation and annihilation
be changed by acting with the creation and annihilation operators for charged particles must transform like the wave
operators. functions of charged particles in ordinary quantum mechan-
ics.15 For electron creation and annihilation operators (taking
B. Metals vs. superconductors the charge of the electron to be e ¼ jej), this means

A metal, like most systems, has a definite particle number, Wr 7! eiea Wr ; (23)
so the expectation value of any operator which does not map
a sector of the Fock space back onto itself will vanish. For W†r 7! eiea W†r : (24)
example, choosing W†r and Wr to create and annihilate elec-
trons of spin r, consider Consequently, operators which conserve electron number,
such as the number operator W†r Wr , are gauge invariant.
hXN jW# W" jXN i ¼ hXN jXN  2e i ¼ 0; (19) Conversely, operators such as W#W" which do not conserve
electron number are not gauge invariant, and transform as
where jXN i is the ground state of the metal: perhaps a filled
Fermi sea. Expectation values such as Eq. (19) are called W# W" 7! e2iea W# W" : (25)
ground state expectation values, correlation functions, or, in
particle physics, vacuum expectation values (VEV’s) and are The expectation value, typically a real quantity, must then
of central importance in field theories, as suggested by their also transform as
many names (which we will use interchangeably). A com-
mon shorthand is to abbreviate these correlation functions as hW# W" i 7! he2iea W# W" i ¼ e2iea hW# W" i: (26)
hW# W" i with the ground states implicit. Additionally, corre-
lation functions such as hW# W" i in Eq. (19) which map a So, for our definition of the order parameter in Eq. (22) to
state to a different sector of Fock space are often called off- make sense, the order parameter itself must also transform as
diagonal, and as we saw vanish for any state of definite parti-
cle number.11 D 7! e2iea D: (27)
Now, let us consider a superconductor, where a new
ground state jXS i has developed. The pair condensate can be Thus, the order parameter is complex, and, more impor-
thought of as the superconducting vacuum (since paired elec- tantly, non-gauge invariant. Since all observable quantities
trons have minimal energy), which we denote as the field must be gauge invariant, this implies the order parameter is
D(x). The electrons in the superconductor may interact pair- not observable. (The magnitude of the order parameter jDj,
wise with the vacuum, either vanishing and creating a pair or the energy gap in the superconductor’s spectrum, is observ-
appearing and breaking a pair, and as a result electron num- able; however its phase is not).
ber is no longer conserved12 and states of different electron Note that the expectation value hWWi is non-gauge invari-
numbers may now have a nonzero overlap. ant in any system, not just a superconductor. However, such
In particular, an expectation value is identically zero in an ordinary sys-
tem, and zero transforms trivially under a gauge transforma-
hXS jW# W" jXS i ¼ hXS jXS  2e þ pairi 6¼ 0: (20) tion. What makes a superconductor special is that this
expectation value is nonzero, allowing its transformation
The ground state may now begin with N electrons, interact properties to be physically relevant.
with the condensate and exchange any number of pairs and
electrons, and then be found in a state with M electrons. V. THE ROLE OF MEAN FIELD THEORY
Now, off-diagonal correlation functions such as Eq. (20) no
longer vanish and superconductors are said to possess off- In Sec. IV, we have seen that the superconducting order
diagonal long range order (ODLRO). This electron number parameter is a gauge covariant quantity, which is to say that
non-conservation is nowhere better illustrated than in the cel- it transforms in a simple—but nontrivial—manner. In Secs.
ebrated BCS wave function for the ground state of a VI–VIII, we will consider descriptions of superconductors
superconductor13 which take D as the primary dynamical quantity, and its

438 Am. J. Phys., Vol. 87, No. 6, June 2019 Nicholas R. Poniatowski 438
gauge covariance will be of central importance. As such, it is invariant, given the transformations from Eqs. (25) and (27).
worth briefly discussing how it is possible for D to attain a Although the mean field approximation is implicit in both
nonzero expectation value, which is the origin all of the the BCS and Ginzburg-Landau theories of superconductivity
physics to follow. that form the cornerstones of the discipline, it is worth noting
The microscopic theory of superconductivity is written in that non-mean field and manifestly gauge invariant
the language of quantum field theory, and the methods nec- approaches to superconductivity exist such as the perturba-
essary for a quantitative treatment of this issue lie well out- tive formalism developed by Nambu,24 as well as electron
side the scope of this article. Here, we will sketch how such number conserving theories such as number projected
an analysis proceeds, and refer the reader seeking further BCS.16,17
detail to the literature.15,21
A quantum field theory can be constructed from a quan- VI. GINZBURG-LANDAU ANALYSIS
tum action, which (roughly speaking) specifies the energy
associated with each allowed microscopic process.22 In a Having seen how it is possible for the order parameter D
superconductor, one such process is the pairing interaction, to acquire a nonzero expectation value in the superconduct-
which contributes a term ing state, let us consider the textbook treatment of symmetry
breaking in superconductors: the Ginzburg-Landau theory,18
in which the free energy of the system is expanded as a series
Spair ¼ g W†" W†# W# W" ; (28) in D near the phase transition where D is small, such that we
need only to keep the first few terms19
where W†r and Wr are the electron creation and annihilation ð 
operators introduced in Sec. IV and g is the electron-phonon 1
F ¼ d3 x ðr þ 2ieAÞD?  ðr  2ieAÞD
coupling constant.21 Unfortunately, such a quartic term in 2m?
the action typically makes the quantum theory not analyti- 
? u ? 2 1 2
cally solvable. So, to progress we must approximate. In par- þrD D þ ðD DÞ þ B þ    : (32)
ticular, we will make a mean field approximation, which 2 2
capitalizes on the macroscopic, semi-classical nature of a
Here, r and u are unknown temperature dependent expansion
superconductor.23
coefficients, and the coefficient of the first term is conven-
To begin, we write a pair of electron annihilation opera-
tionally chosen to be 1/2m?, where m? ¼ 2me is the mass of a
tors as (in what follows we will omit spin indices for
Cooper pair. To ensure the free energy is gauge invariant
brevity)
(and thus observable), the expansion includes only powers of
WW ¼ hWWi þ WW  hWWi  hWWi þ g: (29) the gauge invariant quantity D?D, and in the “kinetic” term
we have again replaced r 7! r  2ieA (since the charge of
In the second equality, we interpret the operator WW as a Cooper pair is 2e). Finally, the B2 term accounts for the
being the expectation value hWWi plus quantum fluctuations, energy of an external magnetic field.
g  WW  hWWi around it. Note, crucially, that WW is an The ground state may then be identified via a variational
operator which annihilates two electrons, whereas its expec- principle: it will be the state for which the free energy is min-
tation value is simply a number. imized with respect to the order parameter. Since spatial var-
Substituting Eq. (29) and a similar expression for W†W† iations of the order parameter are energetically costly, we
into Eq. (28) gives may assume D is homogenous in the ground state of the sys-
tem, and thus the first term in Eq. (32) vanishes. Carrying
Spair ¼ ghW† W† ihWWi  ghW† W† iWW  gW† W† hWWi out the variation imposes the condition
þ Oðg† gÞ: (30) dF
0¼ ¼ ½r þ uðD? DÞD; (33)
dD?
For a macroscopic system such as a superconductor, we may
assume quantum fluctuations are in some sense small, and which has two solutions: D ¼ 0, which we associate with the
neglect terms of order g†g. Defining the order parameter metallic phase above the phase transition, or D?D ¼ –r/u,
D  ghWWi, we are left with which we associate with the superconducting phase.20 Since
this fixes only the amplitude of D, we appear to have a con-
D? D tinuum of ground states, parameterized by the phase angle /,
Spair ¼  D? WW  W† W† D: (31)
g rffiffiffiffiffi
jrj i/
Since the order parameter field D is a number—not an opera- D¼ e : (34)
u
tor—the second two terms in Eq. (31) do not conserve elec-
tron number. This electron non-conservation allows the At first glance, we appear to have a degenerate manifold of
order parameter hWWi to acquire a nonzero vacuum expecta- non-gauge invariant ground states, and are tempted to con-
tion value, since the mean field action now connects sectors clude that gauge symmetry has been spontaneously broken.
of the Fock space with different particle number via the last However, as we will see, this apparent breaking of gauge
two terms. symmetry is merely an illusion, whereby gauge redundancy
Thus, it is ultimately the mean field approximation which and the covariance of the order parameter have conspired to
is responsible for electron number non-conservation, and make a single physical ground state appear to be an infinite
consequently D acquiring a nonzero vacuum expectation set of distinct states. In fact, it is this very deception that is
value. However, notice that the action is still perfectly gauge the origin of the misleading term “broken gauge symmetry.”

439 Am. J. Phys., Vol. 87, No. 6, June 2019 Nicholas R. Poniatowski 439
After getting better acquainted with the subtleties of gauge Al 7! Al þ @ la, so the covariant derivative of the order
invariance in the superconducting state in Sec. VII, we will parameter transforms as
return to this state of affairs in Sec. VIII, and discover what
it really means to “break” a gauge symmetry. Dl D ¼ ð@l  2ieAl ÞD 7! e2iea ð@l þ 2ie @l a
 2ieðAl þ @l aÞÞD
VII. EFFECTIVE FIELD THEORIES
¼ e2iea Dl D: (36)
In general, an effective field theory such as the Ginzburg-
Landau theory discussed above involves trading the micro-
One can easily show that the complex conjugate (DlD)?
scopic Hamiltonian of a system for an effective description
transforms as ðDl DÞ 7! e2iea ðDl DÞ? , so we can form gauge
that captures the low energy physics using emergent degrees
invariant terms from the product (DlD)?DlD. Thus, gauge
of freedom,8 in this case the order parameter field D(x, t),
invariance constrains the effective action to be a functional
which we have now generalized to be time dependent. To
of the form F ½D? D; ðDl DÞ? Dl D.
construct an effective field theory, we write down a
In fact, we can be slightly less restrictive by re-
Lagrangian, Hamiltonian, Free Energy, etc., based on sym-
metry arguments, and (classically) derive the resulting phys- parameterizing the two degrees of freedom of the complex
ics from the appropriate variational procedure. This central order parameter as an amplitude and a phase, D ¼ jDjei/ .
quantity, which we will call F and refer to as the “effective Evidently, any functional of the amplitude will be gauge
action” is generally a functional of the order parameter and invariant. Expressed in terms of the amplitude and phase
its derivatives, and must be gauge invariant (this is one of the fields, the derivatives can be written
symmetries we must take into account). In this section, we  
  @l jDj
will show that the constraint that F be gauge invariant cou- @l D ¼ @l jDjei/ ¼ i@l / þ D: (37)
pled with the gauge-covariance of the order parameter is jDj
enough to account for several of the striking phenomena of
the superconducting state, independent of the choice of for- Recalling our discussion in Sec. VI, the amplitude of the
malism: it doesn’t matter whether F is a Lagrangian or a order parameter will be fixed at the non-zero value which
Free Energy. For the most part, this portion of the paper is a minimizes the system’s energy, and fluctuations around this
more elementary presentation of ideas originally developed minimum will be gapped and energetically costly. Since we
in Refs. 1 and 2. Afterward, we will consider the central role are concerned with the low energy behavior, it is safe to
of the Anderson-Higgs mechanism in the story of gauge assume that @l jDj jDj, and we may drop the second term
symmetry breaking. in Eq. (37) so @ lD
i@ l/D. Note that under a gauge trans-
Since the electromagnetic field, to which the superconduc- formation D 7! e2iea D, and thus the phase transforms as
tor is coupled, is intrinsically relativistic, it is convenient to / 7! / þ 2ea.
adopt relativistic language. Throughout, we will use the met- In this representation, the gauge covariant derivative can
ric with signature (þ, –, –, –), which is the standard in parti- be written as Dl D ¼ ið@l /  2eAl ÞD, and we may recognize
cle physics, and sums over repeated indices are always the term in parentheses to be gauge invariant: @l /  2eAl
implied. Spacetime coordinates are written as xl ¼ (t, x) 7! @l ð/ þ 2eaÞ  2eðAl þ @l aÞ ¼ @l /  2eAl . So, the
where l ¼ 0, 1, 2, 3, derivatives with respect to xl are writ- effective action can be a functional of only the amplitude jDj
ten as @ l, and the scalar and vector potentials are combined and the gauge invariant combination @ l/ – 2eAl such that
into the four-potential Al ¼ (u, –A). Throughout the remain- we have F ½jDj; @l /  2eAl . In fact, there are no other
der of this section, we will make use of this relativistic nota- gauge-invariant combinations of the fields that we could
tion, but at no point will we actually assume the theory we write down, and the previous functional form we considered
are discussing is Lorentz invariant: we are merely using this is a special case of this structure. Thus, we have determined
notation as a convenient shorthand in discussing a presum- the most general constraints that gauge invariance imposes
ably non-relativistic system. The reader unfamiliar with elec- on the effective action.
tromagnetism from the perspective of classical field theory is Now, given the effective action we may determine the
encouraged to consult the literature.3,25,26 ground state by the appropriate variational principle. By
In general, the effective action may depend on the order assumption, the ground state is attained for a non-zero value
parameter field D, its complex conjugate D?, and their deriv- of jDj, which as we saw in Sec. VI is the origin of “broken”
atives @ lD, @ lD?, as well as the electromagnetic field via gauge symmetry. As far as the dependence on @ l/ – 2eAl,
dependences on the potential Al and its derivatives @ lA . So, stability requires that the minimum of F occur for a finite
we expect to have some functional of these quantities, value of its arguments, and we further expect it to be a qua-
F ½D; D? ; @l D; @l D? ; Al ; @l A . However, gauge invariance dratic function near the minimum27
strongly constrains the form of this functional. Clearly the
product D?D is gauge invariant and we may use it to con- F  Kð@l /  2eAl Þ2 ; (38)
struct terms in the effective action, but the derivatives of the
order parameter transform as for some prefactor K. Then, F will be minimized when

@l D 7! @l ðe2iea DÞ ¼ e2iea ð@l þ 2ie@l aÞD; (35) @l / ¼ 2eAl : (39)

which is clearly not gauge invariant. To construct gauge This condition on the ground state follows directly from the
invariant derivative terms, we must introduce the gauge requirement that the effective action be gauge invariant, and
covariant derivative, Dl  @ l – 2ieAl. In our relativistic it is sufficient to derive several of the properties of the super-
notation, a gauge transformation of the potentials is written conducting state.

440 Am. J. Phys., Vol. 87, No. 6, June 2019 Nicholas R. Poniatowski 440
A. Perfect conductivity The existence of an infinite set of ground states, each with a
different phase / led us to believe that gauge symmetry was
The timelike component of Eq. (39) requires that inside a spontaneously broken.
superconductor @ t/ ¼ 2eu, where u is the scalar potential. However, we must not forget that the order parameter
Then, for any time independent field configuration for which does not completely specify the state of the system, we also
@ t/ ¼ 0, we must have have the electromagnetic field. Having seen that the poten-
tials Al are directly tied to the order parameter, and are in
u ¼ 0: (40)
some sense the fundamental entity, we will use Al to specify
the electromagnetic field configuration. We can recover E
So, the voltage between any two points inside the supercon-
and B from the components of the field strength tensor,26
ductor is V ¼ uðr2 Þ  uðr1 Þ ¼ 0. One time independent
Fl ¼ @l A  @ Al , as Ei ¼ F0i and Bi ¼ 12 eijk Fjk , where the
field configuration is a steady current j ¼ const, which by the
roman indices run over the spatial components 1, 2, 3. In the
above argument must occur with zero potential difference. A
finite current maintained at zero voltage implies the conduc- ground state, we have E ¼ B ¼ 0 when Fl ¼ 0, which is
tivity is infinite (since j ¼ rV with V ¼ 0 and j 6¼ 0). Thus, achieved when the potential is a “pure gauge,” Al ¼ @ lb, for
superconductors may maintain persistent, dissipationless any scalar field b. This is simply to say that the potential is
currents.2 related to Al ¼ 0 by a gauge transformation, and thus repre-
sents the same physical state. The important point is that any
choice of b is valid, and thus there are an infinite number of
B. Flux quantization
potentials which describe the same physical state of E ¼ B
Consider a thick superconducting ring, and a closed con- ¼ 0. The ground state of the superconducting system is then
tour c traversing it. The magnetic flux U through the surface given by
S bounded by the contour is
D ¼ D0 ei/ ; Al ¼ @l b: (44)
U ¼ 兼S B  dS
Now, suppose we perform a gauge transformation on this
¼ 兼S ðr  AÞ  dS state, under which the potential transforms Al 7! Al þ @l a,
þ
and the phase of the order parameter transforms as / 7! /
¼ A  ds þ 2e a. The ground state in Eq. (44) then becomes
c
þ
1 D ¼ D0 eið/þ2eaÞ ; Al ¼ @l ða þ bÞ: (45)
¼ r/  ds
2e c
Evidently, by performing a gauge transformation we may
1 rotate the phase of D. Further, since our choice of a is arbi-
¼ D/; (41)
2e trary, we may rotate the phase into any value we wish.
Specifically, we can perform a gauge transformation to move
where we used Stoke’s theorem to get from the second from one ground state of fixed phase to another. We also
equality to the third, and Eq. (39) to get from the third to the know that two field configurations related by a gauge trans-
fourth. The final equality comes from the fact that the closed formation represent the same physical state, and thus con-
line integral over the gradient of / is simply the difference clude that the all of the seemingly degenerate ground states
between / at the beginning and end of the path. Normally, in Eq. (44) are merely different descriptions of the same
such an integral must be zero, but since / is an angle defined physical state. This is the essence of “broken” gauge symme-
only modulo 2p, we may have D/ ¼ 2pn for any integer n. try: there is only one physical ground state, wherein jDj
Thus, the flux through the loop is quantized as ¼ D0 and E ¼ B ¼ 0, but an infinite number of gauge equiv-
alent ways to describe it. The existence of multiple order
hpn
 parameter configurations to describe the same ground state
U¼ : (42) looks extremely similar to the ground state degeneracy aris-
e
ing from a spontaneously broken symmetry, which has led to
VIII. THE MEISSNER EFFECT AND THE the misleading terminology of “spontaneously broken gauge
ANDERSON-HIGGS MECHANISM symmetry.” Crucially, one must note that at no point did
gauge symmetry ever actually “break:” one can verify that
One could use the spatial components of Eq. (39) to show every step of our analysis (and all of our discussion to fol-
B ¼ r  A ¼ r  r/ ¼ 0, which is to say that a magnetic low) is perfectly gauge invariant, as any consistent physical
field cannot exist in the ground state of a superconductor. theory must be. In fact, not only is the violation of gauge
However, while valid, this overlooks the rich physics under- symmetry nonsensical, but, at least in lattice gauge theories,
lying the Meissner effect, namely, the Anderson-Higgs it is not possible for such a symmetry to be spontaneously
mechanism. In fact, this mechanism is the core of the gauge broken due to the famous Elitzur theorem.28
symmetry breaking story, as we will presently see. Inseparable from this illusion of gauge symmetry breaking
Recall the simple Ginzburg-Landau theory from Sec. VI is the Anderson-Higgs mechanism, by which the spectrum of
where we found the ground state of the superconductor the gauge field acquires a gap. In the particle physics vernacu-
is characterized by a nonzero order parameter amplitude, lar, this corresponds to the gauge field acquiring a mass, and
jDj ¼ D0 and the absence of electromagnetic fields. The as this language has bled into the condensed matter literature,
ground state configuration is then given by it is often said that the “photon acquires a mass.”
To illustrate the Anderson-Higgs mechanism in its most
D ¼ D0 ei/ ; E ¼ B ¼ 0: (43) natural context, let us consider the Lorentz invariant

441 Am. J. Phys., Vol. 87, No. 6, June 2019 Nicholas R. Poniatowski 441
analogue of a superconductor: the Abelian-Higgs model, This is the Klein-Gordon equation, which describes a classi-
defined by the Lagrangian density26 cal field of mass m. This is made clear by Fourier transform-
ing the field
 1 ð
L ¼ ðDl DÞ Dl D  Fl Fl  V ðjDjÞ: (46)
4 ~ l
A l ðxÞ ¼ d4 x eipl p A~l ðpÞ; (57)
Although this model does not actually describe a supercon-
ductor, nor does it play any role in the Standard Model, it is where pl ¼ (E, p). Then, the Klein-Gordon equation
an incredibly useful pedagogical tool to understand the becomes
Anderson-Higgs mechanism, especially as it applies to
superconductors. We assume the potential V is minimized by ðpl pl þ m2 ÞA~l ðpÞ ¼ 0; (58)
a nonzero value of the order parameter amplitude, giving
rise to the gauge-equivalent ground states in Eq. (44) just as pl pl þ m2 ¼ 0; (59)
in the non-relativistic case. By parameterizing the order
parameter by its amplitude and phase, the covariant deriva- E2 ¼ p2 þ m2 ; (60)
tive can be written Dl D ¼ ð@l /  2eAl ÞD, and thus the
Lagrangian becomes which we recognize as the relativistic dispersion of a particle
with mass m. If we restrict ourselves to considering only the
1 spacelike components of Eq. (56) (due to the additional
L ¼ jDj2 ð@l /  2eAl Þ2  Fl Fl  V; (47) structure of Lorentz invariance, the timelike component will
4
be substantially different from the nonrelativistic case, to
which we may again rewrite by defining a new vector field which we will momentarily compare our results), and further
specialize to time-independent scenarios, we have
1
A~l  Al  @l /: (48) ~ ¼ 0:
2e ðr2 þ m2 ÞA (61)

Note that this is not a gauge transformation: A~l is a new ~ ¼B

Taking the curl of this equation and noting that r  A
gauge invariant vector field, since under a gauge since r  (r/) ¼ 0, we find an equation for the magnetic
transformation field
1 1 ðr2 þ m2 ÞB ¼ 0; (62)
A~l 7! Al þ @l a  @l ð/ þ 2e aÞ ¼ Al  @l / ¼ A~l :
2e 2e
(49) which notably does not admit a constant solution, implying a
uniform magnetic field cannot exist inside the superconduc-
Noting that Fl is invariant under this redefinition, the tor. In fact, at the interface between a superconductor and
Lagrangian becomes the vacuum, we expect the field to decay exponentially like
emr , which is strongly reminiscent of the Meissner effect.
2 l 1 To be more physically accurate, let us return to supercon-
L ¼ ð2eÞ jDj2 A~l A~  Fl Fl  V: (50)
4 ductivity from the perspective of our original, non-
relativistic Ginzburg-Landau free energy from Eq. (32),
In the field theory literature, the term quadratic in A~l implies which we may rewrite as
this vector field has a mass of m2 ¼ 2e2 jDj2 . To make this
explicit, we begin by finding the Euler-Lagrange equations ð
jDj2 1
of motion for the system F ¼ d3 x ?
ðr/  2eAÞ2 þ B2 þ V ðjDjÞ; (63)
2m 2
@L @L ~
  @l
¼ 0; (51) or, in terms of the vector field A,
~
@A 
@ @l A~ ð
ð2eÞ2 jDj2 2 1 2
F ¼ d3 x ~ þ B þ V ðjDjÞ:
A (64)
m2 A~ þ @ l Fl ¼ 0; (52) 2m? 2
m2 A~ þ @ l ð@l A~  @ A~l Þ ¼ 0; (53) Similarly to the relativistic case, the current is given by14
m2 A~ þ @ l @l A~  @ @ l A~l ¼ 0: (54) dF
2
ð2eÞ jDj2
j¼ ¼ ~
A; (65)
~
dA m?
To simplify this expression, we note the current is given by25
@L and by Maxwell’s equation r  B ¼ j, we have
2~
jl ¼  l ¼ m A l : (55)
@ A~ ð2eÞ2 jDj2
rB¼ ~
A: (66)
m?
For this current to be conserved, we must have @ ljl ¼ 0, and
thus @ l A~l ¼ 0, which requires the last term in Eq. (54) to Taking the curl of both sides, and using r B ¼ 0, we find
vanish. The equation of motion is then simply
ð2eÞ2 jDj2
ð@l @ þ m ÞA~l ¼ 0:
l 2
(56) r2 B ¼  B; (67)
m?

442 Am. J. Phys., Vol. 87, No. 6, June 2019 Nicholas R. Poniatowski 442
 2
from which we again see that a uniform magnetic field is not Steven Weinberg, “Superconductivity for particular theorists,” Prog.
admitted inside a superconductor. If we consider a supercon- Theor. Phys. Suppl. 86(1), 43–53 (1986).
3
ductor occupying the half space x > 0 and vacuum in x < 0, JohnDavid Jackson, Classical Electrodynamics, 2nd ed. (John Wiley &
Sons, New York, NY, 1975), pp. 547–556.
the magnetic field will be given by 4
This is because the Lagrangian has a term qA  x_ coupling the particle to
the field. The canonical momentum p ¼ @L=@ x_ then has an additional term
ð2eÞ2 jDj2  qA which must be subtracted out to get the “physical” momentum mx. _
B ¼ B0 ex=k ; k2 ¼ ; (68) 5
L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic
m?
Theory (Pergamon Press, London, UK, 1958), pp. 471–474.
6
Since a(x, t) is dependent on space and time, derivatives acting on the gauge
where the parameter k is the London Penetration Depth,
transformed wave-functions give rðeiqa wÞ ¼ eiqa rw þ iqðraÞeiqa w, and
which in SI units is similarly for time derivatives.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7
Lewis H. Ryder, Quantum Field Theory, 2nd ed. (Cambridge U.P.,
me Cambridge, UK, 1996), pp. 282–293.
k¼ : (69) 8
2l0 e2 jDj2 Mehran Kardar, Statistical Physics of Fields, 1st ed. (Cambridge U.P.,
Cambridge, UK, 2007), pp. 19–31.
9
Alexander Altland and Ben Simons, Condensed Matter Field Theory, 2nd
Notice that this is completely analogous to the relativistic ed. (Cambridge U.P., Cambridge, UK, 2010), pp. 43–50.
theory, with the role of the dynamically generated 10
Since for any system of definite particle number the energy eigenstates
mass being played by the inverse penetration depth. The will be simultaneous eigenstates of the number operator, there is no way
Anderson-Higgs mechanism sketched here was initially put the system can time evolve into a state with different particle number. Put
forth in the context of superconductivity by Anderson,29 another way, since the number operator commutes with the Hamiltonian,
but has risen to fame in particle physics, wherein this same particle number must be conserved.
11
James F. Annett, Superconductivity, Superfluids and Condensates, 1st ed.
mechanism generates the masses of the W and Z bosons.30 (Oxford U.P., New York, NY, 2004), pp. 106–108.
However, due to the non-Abelian nature of the “broken” 12
Even though the pairs in the condensate are comprised of electrons, we
gauge group in the electroweak theory, some aspects of the treat them as different objects. To be more precise, the electrons we speak
dynamics differ from the case we have discussed here. of are quasiparticles, not bare electrons.
13
Further, just as the Higgs boson was recently discovered at J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of super-
the Large Hadron Collider, there have also been experimen- conductivity,” Phys. Rev. 108, 1175–1204 (1957).
14
Piers Coleman, Introduction to Many-Body Physics, 1st ed. (Cambridge
tal observations of Higgs modes in superconductors.31
U.P., Cambridge, UK, 2015), pp. 372–374.
15
Reference 14, pp. 505–560.
16
Klaus Dietrich, Hans J. Mang, and Jean H. Pradal, “Conservation of parti-
IX. CONCLUSION cle number in the nuclear pairing model,” Phys. Rev. 135, B22–B34
(1964).
Within the mean field approximation, the formation of 17
Yukihisa Nogami, “Improved superconductivity approximation for the
Cooper pairs enables gauge covariant operators such as the pairing interaction in nuclei,” Phys. Rev. 134, B313–B321 (1964).
order parameter D  hWWi to acquire a nonzero expectation 18
L. D. Landau, “On the theory of superconductivity,” in Collected Papers
value in the ground state of the system. The order parameter of L.D. Landau, edited by D. Ter Haar (Gordon and Breach, New York,
amplitude being nonzero in the ground state allows the phase NY, 1965), pp. 546–568.
19
Expanding the free energy, which is known to be discontinuous at a phase
/ to be well-defined but arbitrary, which in turn leads to the
transition, as an analytic series near the discontinuity can only be justified
apparent existence of a degenerate ground state manifold. if we understand the free energy as the result of a saddle point approxima-
However, due to the coupling with the electromagnetic field, tion of the appropriate partition function, see Ref. 8.
all of these ground state solutions are in fact gauge equiva- 20
Details of the Ginzburg-Landau theory, which we have glossed over,
lent descriptions of the same physical state. The excitations ensure that in the superconducting state r < 0, so the gap magnitude D†D
of the gauge field away from this ground state are gapped ¼ –r/u is a positive quantity, see Ref. 14.
21
due to the Anderson-Higgs mechanism, which is responsible Reference 9, pp. 281–316.
22
R. Shankar, Quantum Field Theory and Condensed Matter, 1st ed.
for the Meissner effect. In fact, we have seen that the gauge (Cambridge U.P., Cambridge, UK, 2017), pp. 52–104.
principal is sufficient to account for many of the striking phe- 23
Henrik Bruus and Karsten Flensberg, Introduction to Many-Body
nomena observed in superconductors. In all, the question of Quantum Theory in Condensed Matter Physics, 1st ed. (Oxford U.P.,
whether or not gauge symmetry “breaks” in a superconduc- Oxford, UK, 2004), pp. 65–85.
24
tor is one of linguistics, but it is unambiguous that gauge Yoichiro Nambu, “Quasi-particles and gauge invariance in the theory of
invariance is absolutely central to the physics of superconductivity,” Phys. Rev. 117, 648–663 (1960).
25
Matthew D. Schwartz, Quantum Field Theory and the Standard Model
superconductors. (Cambridge U.P., Cambridge, UK, 2014), pp. 49–65.
26
Valery Rubakov, Classical Theory of Gauge Fields, 1st ed. (Princeton
U.P., Princeton, NJ, 2002), pp. 3–8.
ACKNOWLEDGMENTS 27
Once could conceivably ask why the quadratic function must be centered about
zero, i.e., why could we not have something like
The author would like to thank Tom Cohen, Chris Lobb, F  K½ð@l /  2eAl Þ2  j2 2 ? The effective action would then be minimized
Rick Greene, and Sankar Das Sarma for productive when jl ¼ @l /  2eAl , which we know from Eq. (55) represents current
conversations instrumental to clarifying his thinking and flow.
28
providing feedback on the manuscript. This work is partially S. Elitzur, “Impossibility of spontaneously breaking local symmetries,”
supported by the NSF Award No. DMR-1708334 and the Phys. Rev. D 12, 3978–3982 (1975).
29
P. W. Anderson, “Plasmons, gauge invariance, and mass,” Phys. Rev. 130,
Maryland Center for Nanophysics and Advanced Materials 439–442 (1963).
(CNAM). 30
The Anderson-Higgs mechanism also generates the mass of quarks and
electrons due to a Yukawa coupling that has no analogue in superconduc-
a)
Electronic mail: nponiat@umd.edu tors, and thus is not discussed here.
1 31
S. Weinberg, The Quantum Theory of Fields, 1st ed. (Cambridge U.P., M. A. Measson et al., “Amplitude Higgs mode in the 2H-NbSe2 super-
Cambridge, UK, 1995), Vol. II, pp. 295–358. conductor,” Phys. Rev. B 89, 060503 (2014).

443 Am. J. Phys., Vol. 87, No. 6, June 2019 Nicholas R. Poniatowski 443